Lesson 5. Chapter 7. Wiener Filters. Bengt Mandersson. r x k We assume uncorrelated noise v(n). LTH. September 2010

Transcription

1 Optimal Sigal Poceig Leo 5 Chapte 7 Wiee Filte I thi chapte we will ue the model how below. The igal ito the eceive i ( ( iga. Nomally, thi igal i ditubed by additive white oie v(. The ifomatio i i (. Alo, we ofte ued the appoach that the ifomatio igal i modeled a white oie w( filteed i a filte g(. Chapte 7. Wiee Filte w( g ( ( ( oie v( ( h( H( etimated output deied output d( LTH Septembe We will imie the output, which we decibe a the diffeece betwee the deied output d( the etimated output. ˆ imie Ee [ ( ] E[( d ( d ( ] Begt Meo Depatmet of Electical Ifomatio Techology, Lud Uiveity Lud Uiveity Applicatio. Filteig (: Smoothig: Pedictio: Equaliatio: The deied igal i ( we will detee the optimum filte fo oie eductio. Like filteig but we allow a eta delay i the output igal (pecially image poceig. The output i a pedictio of futue value of (. Oe tep pedicto. pedict et value (. The deied igal i w( we will detee the optimum filte fo whiteig the output pectum (ivee filteig, decovolutio. Othe applicatio: Echo cacellatio. Noie cacellatio. Pule hapig Pedictio Eo Filte PEF (ecod ode fom chapte 4 Optimum Filte (poce the igal ( w( δ ( Model of the igal (. Iput: white oie w ( o impule δ ( A ( ( ( k A A ( a( a( v( w( ( ( g ( δ ( k We aume ucoelated oie v(. h ( d$( d( ( ˆ( l l a ( ( ( ˆ( a ( ( d ( could be: (, filteig oiy igal ( (, moothig (allow delay (, pedict futue value δ (, ivee filteig, decovolutio We ca ewite the figue w( δ ( - A d ( dˆ( ˆ( ( ˆ( d ( ( h( [ a, a ] d$( $( deied igal ( ökad etimated igal igal h( caual FIR filte: eay, ueful (chap. 7. h( ocaual IIR filte: eay, le ueful (chap h( caual IIR filte moe difficult, ueful (chap We aume that coelatio fuctio, d ae kow o could be etimated

2 Deivatio of the optimal olutio (Wiee filte. Real-valued om igal. We tat ˆ Ee [ ( ] E[( d ( d ( ] d ˆ( hl ( ( l (i geeal ocaual filte l e ( d ( d ˆ( d ( hl ( ( l Set the deivative of epect to h equal to eo fo all k. ( Ee Ee hk hk ( e Ee k which give [ ( ] [ ( ( ] [ ( ( ( ] Ee [( ( ] (The othogoality piciple Replace the E[( d( h( ( ( ] l Deivatio of the imum Witig [ ( ] [ ( ( ] { ( [ ( ( ( ]} l Ee Eee Ee d hl l Eed {( ( hl ( Ee {( ( } l give data othogoal Thi give the imum Eed [ ( ( ] E{[ d( hl ( ( ] d( } l ( h( l ( ( h( l ( l d d d l l d( h( ( l we got the Wiee-Hopf equatio l hl ( ( k The Wiee filte wa deive fom om igal. Fo a deteitic appoach we have to ue the defiitio of autocoelatio co coelatio The, imie ( ( d( ( ˆ ( d ( d ( ε The Wiee-Hopf equatio will be the ame. FIR Wiee filte (pp , table 7. page 339 The Wiee-Hopf equatio ae ow p l o i mati fom hl ( ( k k,,..., p ( ( ( ( p h( ( ( ( ( ( p h( ( ( ( ( ( p 3 h( (. ( p ( p ( p 3 ( hp ( ( p 443 R h R h Now, we will look at the thee type of filte H( FIR Wiee filte (i the tetbook deoted W( Nocaual IIR filte Caual Wiee filte (at the ed of thi chapte The olutio i the imum which alo ca be witte h R p d l ( h( ( p T d ( h( ( d ( hopt d ( l T R 79 8

3 Nocaual IIR Wiee filte (pp , table 7. The Wiee-Hopf equatio ae hee l hl ( ( k all k Filteig igal fo oie eductio The igal i ditubed by additive eo mea white oie ( ( v( Hee we have a complete covolutio it ca be olved uig -tafom o Fouie tafom H ( P P P H ; P j ω P He ( j ω P The imum i d( h( ( l w( g ( ( Deied igal i ow (. The v E[ d( ( ] E[ ( ( ( v( ] P P Pv P P ( oie v( ( h( H( Etimated deied d(( We ca ue the Paeval elatio alo wite thi i the fequecy domai. The, (ee popetie of the Fouie tafom, ee page 356, Table 7. π π π [ P H P ( e ] dω d 8 8 Caual FIR-filte fo oie eductio The FIR-filte equatio ae p l Now, they will be p o l hl ( ( k k,,..., p h(( l ( k ( k v ( R R h v h ( R R opt v The pectum we fid fom the Fouie Tafom H Fouie{ h ( } opt Nocaual IIR-filte fo oie eductio Fo o-caual IIR filte, we have H ( P P P H ; P P He ( j ω P I the filteig poblem the powe pecta ae P P P v P P which give the Wiee filte P H ; P P P He ( ; P P v v We ee that fo fequecie low oie, H 83 84

4 Pedictio I a oe-tep pedicto, the deied igal i (. Noie cacellatio (page 349 A igal i ditubed by additive oie v (. w( g ( ( oie v( ( ( h( H( etimated deied d(( Ty to meaue the oie v( fom the ouce etimate the oie v ( added to the igal. The ubtact the oie v ( fom the igal. Deied igal i ow (. The ( k E[ d(( ] E[( (( ] ( k P P Thi give the Wiee-Hopf equatio Sigal ouce Noie ouce v( v( ( (v ( ( H( v ( Wiee filte Etimate of v ( p l hl ( ( k ( k k,,..., p Decovolutio (equaliig, ivee filteig Deied igal hee i w( (o allow delay, w(-. (ee alo poblem 4.9 Thi mea that g( h( δ ( Caual IIR Wiee filte (page Deivatio of the caual filte i moe difficult. The Wiee olutio i l h( ( k w( g ( ( ( oie v( ( h( H( etimated deied d(w(- We divide the olutio ito two tep. Step Step w( ( ε( F( /F( H( ( H( I tep, we whiteig the iput igal (. Fom chapte 3, we have pectal factoiatio P If the choe F( vaiace equal to. the igal ε( will be white 87 88

5 IIR, caual filte Step I tep we kow have (Wiee-Hopf equatio l g( ε ( k ε δ IIR, caual filte 3 Vi have to detee P (. The whee E{ d( ε( E{ d( ( l l f ( f ( Z f ( ( k } ( k { F( } The The optimal filte (the caual filte, k i the the -tafom g u( [ P ( ] The otatio [ ] mea the caual pat of the agumet. P P( P( F( To fid ( we take the caual pat ( P Combiig tep tep give fially H F P ( ( ( 89 9 Relatio betwee caual o caual IIR Wiee filte No caual IIR Wiee filte P( H P P( ( Caual IIR Wiee filte H P ( We ca ee both filte a a cacade two filte thee the fit i a whiteig filte. The imum i a befoe d l ( h( ( Adaptive filteig. Chapte 9 o the coue Adaptive Sigal Poceig. We wat to imie the Ee [ ( ] E[( d ( d ˆ( ] Iteative olutio We ca olve thi iteatively uig the update equatio h ( h δ μ' δ h h μ' E{ ( thee μ' i the tep ie. Adaptive olutio (Leat Mea Squae, LMS Ue the appoimatio E{ ( ( which give h ( h ' ( μ μ '? How to choe tep ie Doe the algoithm covege? How fat? 9 9